ON GENERALIZED STIRLING NUMBERS AND POLYNOMIALS

Singh Chandel introduced the following generalizations of the Stirling numbersand polynomials (see [15]):(5)and(6)whereS (α,λ) (n, k, r) = (−1)kk!T (α,λ)( )k∑ k(−1) j (α + rj) (λ−1,n) ,jj=0nn (x, r, −p) = x n(λ−1) ∑S (α,λ) (n, k, r)p k x rka (λ−1,n) =k=o( ) a(λ − 1) n .λ − 1 nEvidently, when λ → 1, equations (5) and (6) would reduce to (3) and (4)respectively, which, in turns, will yield (1) and (2), respectively for r−1 = α = 0.In this paper we prove that the generalizations (3) and (4) and relatedproperties are an old result, published in 1887. by d’Ocagne ([11]). Also, weprove that Singh Chandel result is the consequence of the fundametal results ofToscano ([19]) and Chak ([6]).The new explicit expressions for numbers (3) and (5) are also given.2. ON A RESULT OF R. P. SINGHOne of the first generalization of the Stirling numbers of the second kind (1)is given by d’Ocagne (see [11]):(7)with the recurrence relationS (α) (n, k) = (−1)kk!( )k∑ k(−1) j (α + j) njj=0(8)S (α) (n, k) = S (α) (n − 1, k − 1) + (k + α)S (α) (n − 1, k)Toscano introduced similar numbers and related polynomials in many papers(see for example [18] and [19] for references).For numbers defined by (7), we have the generating function, recurrencerelations (see [19]):(9)∞∑S (α) (n, k) tnn! = (−1)k e αt (1 − e t ) k ,k!n=02